Finite Group Databasebeta

← C67×#24010 #30703261

Subgroups — C67×#24010 #30703261

Order 17152.

1428 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C67C211yesmaximal, characteristic
2C2×#485×C67C212yesmaximal, complemented
3C67×#2280C211yesmaximal, characteristic
4C67×#2036C212yesmaximal, complemented
5C67×#2035C212yesmaximal, complemented
6C67×#1873C212yesmaximal, complemented
7C67×#1853C211yesmaximal, characteristic, complemented
8C67×#1852C211yesmaximal, characteristic, complemented
9C67×#1787C211yesmaximal, characteristic, complemented
10C67×#1787C211yesmaximal, characteristic, complemented
11C67×#1779C211yesmaximal, characteristic
12C23×Q8×C67C2211yescharacteristic
13C24×C4×C67C2211yescharacteristic
14C2×C22⋊Q8×C67—21no—
15C2×C22⋊Q8×C67—21no—
16C2×C22⋊Q8×C67—28no—
17C2×C22⋊Q8×C67C2212yes—
18C2×C22⋊Q8×C67C2212yes—
19C2×#117×C67C2211yescharacteristic
20C2×#117×C67C2211yescharacteristic
21C22×C4⋊C4×C67—21no—
22C22×C4⋊C4×C67C2211yescharacteristic
23C22×C22⋊C4×C67—21no—
24#485×C67—28no—
25C2×#107×C67C2212yescomplemented
26C2×#106×C67C2212yescomplemented
27C2×#103×C67C2212yescomplemented
28C2×#103×C67C2212yescomplemented
29C2×#103×C67C2212yescomplemented
30C2×#103×C67C2212yescomplemented
31C2×C22⋊C8×C67C2211yescharacteristic
32C2×C22⋊C8×C67C2211yescharacteristic
33C2×C22⋊C8×C67C2212yes—
34C2×C22⋊C8×C67C2212yes—
35#381×C67—21no—
36#380×C67C2211yescharacteristic
37#380×C67C2211yescharacteristic
38#380×C67C2211yescharacteristic
39#380×C67C2211yescharacteristic
40#379×C67—21no—
41C4×C22⋊C4×C67—21no—
42C4×C22⋊C4×C67—21no—
43#340×C67C2212yescomplemented
44#340×C67C2212yescomplemented
45C25×C67D411yescharacteristic
46C22×Q8×C67—24no—
47C22×Q8×C67—21no—
48C22×Q8×C67—22no—
49C22×Q8×C67—22no—
50C22×Q8×C67—42no—
51C22×Q8×C67C2312yes—
52C23×C4×C67—21no—
53C23×C4×C67—21no—
54C23×C4×C67—24no—
55C23×C4×C67—22no—
56C23×C4×C67—22no—
57C23×C4×C67—42no—
58C23×C4×C67D411yescharacteristic
59C23×C4×C67D411yescharacteristic
60C23×C4×C67D411yescharacteristic
61C23×C4×C67C2311yescharacteristic
62C23×C4×C67C2311yescharacteristic
63C23×C4×C67C2311yescharacteristic
64C22×C8×C67C2312yes—
65C22×C8×C67C2312yes—
66C22⋊Q8×C67—28no—
67C22⋊Q8×C67—28no—
68C22⋊Q8×C67—24no—
69C22⋊Q8×C67—24no—
70C22⋊Q8×C67—24no—
71C22⋊Q8×C67D418yes—
72#117×C67—24no—
73#117×C67—42no—
74C2×C4⋊C4×C67—28no—
75C2×C4⋊C4×C67—22no—
76C2×C4⋊C4×C67—22no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—24no—
79C2×C4⋊C4×C67C2311yescharacteristic
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2311yescharacteristic
82C2×C4⋊C4×C67C2311yescharacteristic
83C2×C4⋊C4×C67C2312yes—
84C2×C22⋊C4×C67—28no—
85C2×C22⋊C4×C67—22no—
86C2×C22⋊C4×C67—22no—
87C2×C22⋊C4×C67—21no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67—21no—
90C2×C22⋊C4×C67—21no—
91C2×C22⋊C4×C67—21no—
92C2×C22⋊C4×C67—21no—
93C2×C42×C67—21no—
94C2×C42×C67—21no—
95#107×C67—24no—
96#106×C67—24no—
97#103×C67—24no—
98#103×C67—24no—
99#103×C67—42no—
100#103×C67—42no—
101C22⋊C8×C67—24no—
102C22⋊C8×C67—22no—
103C22⋊C8×C67—22no—
104C22⋊C8×C67—42no—
105#95×C67—21no—
106#95×C67—21no—
107#95×C67—21no—
108#95×C67—21no—
109C24×C67—22no—
110C24×C67—22no—
111C24×C67—22no—
112C24×C67—22no—
113C24×C67—22no—
114C24×C67—41no—
115C24×C67—41no—
116C24×C67C2×D411yescharacteristic
117C24×C67C2×D411yescharacteristic
118C24×C67C2×D411yescharacteristic
119C2×Q8×C67—24no—
120C2×Q8×C67—22no—
121C2×Q8×C67—22no—
122C2×Q8×C67—22no—
123C2×Q8×C67—22no—
124C2×Q8×C67—22no—
125C2×Q8×C67—48no—
126C2×Q8×C67—48no—
127C2×Q8×C67—42no—
128C2×Q8×C67—42no—
129C2×Q8×C67C2×D414yes—
130C22×C4×C67—22no—
131C22×C4×C67—21no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—22no—
135C22×C4×C67—24no—
136C22×C4×C67—24no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—22no—
144C22×C4×C67—22no—
145C22×C4×C67—22no—
146C22×C4×C67—21no—
147C22×C4×C67—21no—
148C22×C4×C67—22no—
149C22×C4×C67—21no—
150C22×C4×C67—21no—
151C22×C4×C67—21no—
152C22×C4×C67—21no—
153C22×C4×C67—41no—
154C22×C4×C67—41no—
155C22×C4×C67—41no—
156C22×C4×C67—48no—
157C22×C4×C67—48no—
158C22×C4×C67—42no—
159C22×C4×C67—42no—
160C22×C4×C67—42no—
161C22×C4×C67—42no—
162C22×C4×C67C2×D411yescharacteristic
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67C2×D414yes—
165C22×C4×C67C2×D411yescharacteristic
166C22×C4×C67#4112yes—
167C22×C4×C67#4112yes—
168C22×C4×C67C2411yescharacteristic
169C2×C8×C67—22no—
170C2×C8×C67—22no—
171C2×C8×C67—22no—
172C2×C8×C67—22no—
173C2×C8×C67—22no—
174C2×C8×C67—22no—
175C4⋊C4×C67—24no—
176C4⋊C4×C67—28no—
177C4⋊C4×C67—28no—
178C4⋊C4×C67—22no—
179C4⋊C4×C67—22no—
180C4⋊C4×C67—22no—
181C4⋊C4×C67—22no—
182C4⋊C4×C67—22no—
183C4⋊C4×C67C2×D414yes—
184C22⋊C4×C67—28no—
185C22⋊C4×C67—28no—
186C22⋊C4×C67—42no—
187C22⋊C4×C67—42no—
188C22⋊C4×C67—42no—
189C22⋊C4×C67—42no—
190C22⋊C4×C67—42no—
191C22⋊C4×C67—42no—
192C42×C67—42no—
193C42×C67—42no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—21no—
197C23×C67—21no—
198C23×C67—28no—
199C23×C67—21no—
200C23×C67—21no—
201C23×C67—21no—
202C23×C67—21no—
203C23×C67—21no—
204C23×C67—21no—
205C23×C67—21no—
206C23×C67—21no—
207C23×C67—21no—
208C23×C67—21no—
209C23×C67—21no—
210C23×C67—21no—
211C23×C67—44no—
212C23×C67—44no—
213C23×C67—44no—
214C23×C67—44no—
215C23×C67—44no—
216C23×C67—44no—
217C23×C67#12012yes—
218C23×C67#12312yes—
219C23×C67#13512yes—
220C23×C67#13712yes—
221C23×C67#13712yes—
222C23×C67C22×D411yescharacteristic
223Q8×C67—24no—
224Q8×C67—24no—
225Q8×C67—48no—
226Q8×C67—42no—
227Q8×C67—42no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—24no—
236C2×C4×C67—24no—
237C2×C4×C67—24no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—21no—
241C2×C4×C67—44no—
242C2×C4×C67—44no—
243C2×C4×C67—44no—
244C2×C4×C67—48no—
245C2×C4×C67—48no—
246C2×C4×C67—42no—
247C2×C4×C67—42no—
248C2×C4×C67—42no—
249C2×C4×C67—42no—
250C2×C4×C67—42no—
251C2×C4×C67—42no—
252C2×C4×C67—41no—
253C2×C4×C67—41no—
254C2×C4×C67—41no—
255C2×C4×C67—41no—
256C2×C4×C67—41no—
257C2×C4×C67—41no—
258C2×C4×C67—41no—
259C2×C4×C67—41no—
260C2×C4×C67—41no—
261C2×C4×C67—41no—
262C2×C4×C67—41no—
263C2×C4×C67—41no—
264C2×C4×C67#12012yes—
265C2×C4×C67#12312yes—
266C2×C4×C67C22×D412yes—
267C2×C4×C67C2×#4111yescharacteristic
268C2×C4×C67C2×#4111yescharacteristic
269C2×C4×C67H(5,𝔽2)12yes—
270C8×C67—42no—
271C8×C67—42no—
272C22×C67—22no—
273C22×C67—22no—
274C22×C67—22no—
275C22×C67—22no—
276C22×C67—22no—
277C22×C67—22no—
278C22×C67—22no—
279C22×C67—22no—
280C22×C67—22no—
281C22×C67—22no—
282C22×C67—22no—
283C22×C67—22no—
284C22×C67—416no—
285C22×C67—41no—
286C22×C67—41no—
287C22×C67—41no—
288C22×C67—41no—
289C22×C67—41no—
290C22×C67—41no—
291C22×C67—41no—
292C22×C67—41no—
293C22×C67#48514yescomplemented
294C22×C67C2×#12011yescharacteristic
295C22×C67C2×#12311yescharacteristic
296C22×C67#53411yescharacteristic
297C22×C67C2×#13511yescharacteristic
298C22×C67C2×#13711yescharacteristic
299C22×C67C2×#13711yescharacteristic
300C22×C67#57511yescharacteristic
301C4×C67—22no—
302C4×C67—22no—
303C4×C67—41no—
304C4×C67—44no—
305C4×C67—44no—
306C4×C67—81no—
307C4×C67—81no—
308C4×C67—81no—
309C4×C67—81no—
310C4×C67#54612yes—
311C4×C67#54612yes—
312#24010C6711yesmaximal, characteristic, direct product factor, Sylow
313C2×C67—24no—
314C2×C67—42no—
315C2×C67—42no—
316C2×C67#240511yescharacteristic
317C2×C67#300211yescharacteristic
318C2×C67#300411yescharacteristic
319C2×C67C2×#48511yescharacteristic
320C2×C67#309511yescharacteristic
321C2×C67#319011yescharacteristic
322C2×C67#319611yescharacteristic
323C22×C22⋊Q8C2×C6711yescharacteristic
324C2×#485C2×C6712yescomplemented
325#2280C2×C6711yescharacteristic
326#2036C2×C6712yescomplemented
327#2035C2×C6712yescomplemented
328#1873C2×C6712yescomplemented
329#1853C2×C6711yescharacteristic, complemented
330#1852C2×C6711yescharacteristic, complemented
331#1787C2×C6711yescharacteristic, complemented
332#1787C2×C6711yescharacteristic, complemented
333#1779C2×C6711yescharacteristic
334C67#2401011yescharacteristic, direct product factor, Sylow
335C23×Q8C22×C6711yescharacteristic
336C24×C4C22×C6711yescharacteristic
337C2×C22⋊Q8—21no—
338C2×C22⋊Q8—21no—
339C2×C22⋊Q8—28no—
340C2×C22⋊Q8C22×C6712yes—
341C2×C22⋊Q8C22×C6712yes—
342C2×#117C22×C6711yescharacteristic
343C2×#117C22×C6711yescharacteristic
344C22×C4⋊C4—21no—
345C22×C4⋊C4C22×C6711yescharacteristic
346C22×C22⋊C4—21no—
347#485—28no—
348C2×#107C22×C6712yescomplemented
349C2×#106C22×C6712yescomplemented
350C2×#103C22×C6712yescomplemented
351C2×#103C22×C6712yescomplemented
352C2×#103C22×C6712yescomplemented
353C2×#103C22×C6712yescomplemented
354C2×C22⋊C8C22×C6712yes—
355C2×C22⋊C8C22×C6711yescharacteristic
356C2×C22⋊C8C22×C6711yescharacteristic
357C2×C22⋊C8C22×C6712yes—
358#381—21no—
359#380C22×C6711yescharacteristic
360#380C22×C6711yescharacteristic
361#380C22×C6711yescharacteristic
362#380C22×C6711yescharacteristic
363#379—21no—
364C4×C22⋊C4—21no—
365C4×C22⋊C4—21no—
366#340C22×C6712yescomplemented
367#340C22×C6712yescomplemented
368C25D4×C6711yescharacteristic
369C22×Q8—24no—
370C22×Q8—21no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8C23×C6712yes—
375C23×C4—21no—
376C23×C4—21no—
377C23×C4—24no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—42no—
381C23×C4D4×C6711yescharacteristic
382C23×C4D4×C6711yescharacteristic
383C23×C4D4×C6711yescharacteristic
384C23×C4C23×C6711yescharacteristic
385C23×C4C23×C6711yescharacteristic
386C23×C4C23×C6711yescharacteristic
387C22×C8C23×C6712yes—
388C22×C8C23×C6712yes—
389C22⋊Q8—28no—
390C22⋊Q8—28no—
391C22⋊Q8—24no—
392C22⋊Q8—24no—
393C22⋊Q8—24no—
394C22⋊Q8D4×C6718yes—
395#117—24no—
396#117—42no—
397C2×C4⋊C4—28no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4C23×C6711yescharacteristic
403C2×C4⋊C4C23×C6711yescharacteristic
404C2×C4⋊C4C23×C6711yescharacteristic
405C2×C4⋊C4C23×C6711yescharacteristic
406C2×C4⋊C4C23×C6712yes—
407C2×C22⋊C4—28no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42—21no—
417C2×C42—21no—
418#107—24no—
419#106—24no—
420#103—24no—
421#103—24no—
422#103—42no—
423#103—42no—
424C22⋊C8—24no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—42no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—41no—
438C24—41no—
439C24C2×D4×C6711yescharacteristic
440C24C2×D4×C6711yescharacteristic
441C24C2×D4×C6711yescharacteristic
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8C2×D4×C6714yes—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4C2×D4×C6711yescharacteristic
486C22×C4C2×D4×C6711yescharacteristic
487C22×C4C2×D4×C6714yes—
488C22×C4C2×D4×C6711yescharacteristic
489C22×C4#41×C6712yes—
490C22×C4#41×C6712yes—
491C22×C4C24×C6711yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4C2×D4×C6714yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C42—42no—
516C42—42no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—28no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23#120×C6712yes—
541C23#123×C6712yes—
542C23#135×C6712yes—
543C23#137×C6712yes—
544C23#137×C6712yes—
545C23C22×D4×C6711yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—48no—
549Q8—42no—
550Q8—42no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4#120×C6712yes—
588C2×C4#123×C6712yes—
589C2×C4C22×D4×C6712yes—
590C2×C4C2×#41×C6711yescharacteristic
591C2×C4C2×#41×C6711yescharacteristic
592C2×C4H(5,𝔽2)×C6712yes—
593C8—42no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22#485×C6714yescomplemented
617C22C2×#120×C6711yescharacteristic
618C22C2×#123×C6711yescharacteristic
619C22#534×C6711yescharacteristic
620C22C2×#135×C6711yescharacteristic
621C22C2×#137×C6711yescharacteristic
622C22C2×#137×C6711yescharacteristic
623C22#575×C6711yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—44no—
628C4—44no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4#546×C6712yes—
634C4#546×C6712yes—
635C2—24no—
636C2—42no—
637C2—42no—
638C2C67×#240511yescharacteristic
639C2C67×#300211yescharacteristic
640C2C67×#300411yescharacteristic
641C2C2×#485×C6711yescharacteristic
642C2C67×#309511yescharacteristic
643C2C67×#319011yescharacteristic
644C2C67×#319611yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC25×C67D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#485×C67C2semidirect
4. C67×#2036C2semidirect
5. C67×#2035C2semidirect
6. C67×#1873C2semidirect
7. C67×#1853C2semidirect
8. C67×#1852C2semidirect
9. C67×#1787C2semidirect
10. C67×#1787C2semidirect
25. C2×#107×C67C22semidirect
26. C2×#106×C67C22semidirect
27. C2×#103×C67C22semidirect
28. C2×#103×C67C22semidirect
29. C2×#103×C67C22semidirect
30. C2×#103×C67C22semidirect
43. #340×C67C22semidirect
44. #340×C67C22semidirect
293. C22×C67#485semidirect
312. #24010C67direct (N × H)
324. C2×#485C2×C67semidirect
326. #2036C2×C67semidirect
327. #2035C2×C67semidirect
328. #1873C2×C67semidirect
329. #1853C2×C67semidirect
330. #1852C2×C67semidirect
331. #1787C2×C67semidirect
332. #1787C2×C67semidirect
334. C67#24010direct (N × H)
348. C2×#107C22×C67semidirect
349. C2×#106C22×C67semidirect
350. C2×#103C22×C67semidirect
351. C2×#103C22×C67semidirect
352. C2×#103C22×C67semidirect
353. C2×#103C22×C67semidirect
366. #340C22×C67semidirect
367. #340C22×C67semidirect
616. C22#485×C67semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C22⋊Q8×C672
C2×#485×C672
C67×#22802
C67×#20362
C67×#20352
C67×#18732
C67×#18532
C67×#18522
C67×#17872
C67×#17792
#2401067